Patch Similarity Modulus and Difference Curvature Based Fourth-Order Partial Differential Equation for Image Denoising

The traditional fourth-order nonlinear diffusion denoising model suffers the isolated speckles and the loss of fine details in the processed image. For this reason, a new fourth-order partial differential equation based on the patch similarity modulus and the difference curvature is proposed for image denoising. First, based on the intensity similarity of neighbor pixels, this paper presents a new edge indicator called patch similarity modulus, which is strongly robust to noise. Furthermore, the difference curvature which can effectively distinguish between edges and noise is incorporated into the denoising algorithm to determine the diffusion process by adaptively adjusting the size of the diffusion coefficient.The experimental results show that the proposed algorithm can not only preserve edges and texture details, but also avoid isolated speckles and staircase effect while filtering out noise. And the proposed algorithm has a better performance for the images with abundant details. Additionally, the subjective visual quality and objective evaluation index of the denoised image obtained by the proposed algorithm are higher than the ones from the related methods.


Introduction
In the field of image processing, image denoising is the basis of image analysis, pattern recognition, and machine vision.It has been widely used in various applications, such as medical images, remote sensing images, and radiographic images.Therefore, image denoising has become a very intensive research topic.
The traditional denoising methods, such as the Gaussian filter, median filter, average filter, and wiener filter, can remove noise well.Unfortunately, these methods are not effective for preserving image edges and texture details.In recent years, novel image denoising methods based on the wavelet transform [1], the nonlocal mean [2], and the partial differential equation (PDE) [3][4][5][6] have been proposed to get clear and high-quality images.Thereinto, nonlinear anisotropic diffusion models based on the PDE are able to effectively relieve the contradiction between noise removal and edge preservation, which motivates the researchers' considerable interest.
In 1990, Perona and Malik (PM) first proposed the classical anisotropic diffusion PDE [7] to improve the isotropic diffusion PDE, which is expressed as where div is the divergence operator and |∇| is the absolute value of the gradient of the image .The diffusion coefficient (|∇|) is a nonnegative function that is given by or where  is the gradient threshold.In 1992, Catté et al. [8] introduced the Gaussian smoothing kernel to regularize 2 Mathematical Problems in Engineering the PM model, which can effectively solve the ill-posed problem of the PM model.Recently, a series of research has been carried out on the PM model, which mainly focuses on how to preserve texture details and avoid staircase effect.In order to solve the problem that the PM model may obscure edges and fine details, the local gray-level variance [9], the local entropy [10], and the difference curvature [11] that can effectively distinguish between fine details and noise were introduced to dynamically change the size of the gradient threshold and then slow the diffusion process in the detail region, achieving the goal of removing noise and retaining more image details.On the other hand, You and Kaveh (YK) introduced a fourth-order PDE-based denoising model [12], in which the absolute value of Laplacian is used as the edge indicator.The YK model can suppress the staircase effect caused by the PM model.Even so, it has the weak capacity of edge preservation and tends to leave the processed image with isolated black and white speckles.So far, much research regarding speckle removal and edge preservation has been devoted to improving the YK model [13][14][15][16][17].For instance, Liu et al. presented an adaptive fourth-order PDE filter [14] for image denoising, which can maintain the jump discontinuities while removing noise; Jidesh and George developed a gauss curvature driven fourthorder diffusion equation [16] adopted in image denoising, which can provide a natural look to the filtered image without causing staircase artifacts; a new PDE combining a second-order filter and a fourth-order filter [17] was proposed by Liu and Xiang, which can improve the quality of the denoised image.Nevertheless, it is difficult for this method to artificially select a series of parameters.
In this paper, we propose a new fourth-order PDE combining a new edge indicator called patch similarity modulus and the difference curvature for image denoising, which takes the superiority of the patch similarity modulus to remove noise and preserves edge features by the inherent characteristics of the difference curvature.Comparative studies with the most relevant image denoising methods in the literature demonstrate that the proposed algorithm can significantly improve the denoised image, and what is more, it can keep more weak edges and details.
The remainder of this paper is organized as follows.Section 2 reviews the related work about fourth-order PDE.Section 3 presents the new edge indicator, patch similarity modulus, and a new fourth-order PDE based on the patch similarity modulus and the difference curvature.Section 4 presents the numerical implementation of the proposed algorithm.The experimental results are demonstrated in Section 5, and the conclusion is drawn from this paper in Section 6.

Related Work about Fourth-Order PDE
It is well known that second-order PDEs, such as the PM model and total variation model, determine the diffusion degree of edge and nonedge regions with the gradients of different directions, which have achieved a good tradeoff between noise removal and edge preservation.However, these methods tend to exhibit staircase effect in the denoised image, which will mislead the postprocessing analysis and interpretation.In order to avoid staircase artifacts, You and Kaveh first proposed a fourth-order PDE for image denoising [12].The energy functional introduced in [12] is the corresponding Euler equation of which can be solved through the following gradient descent procedure: Here, Ω is the image support,  is the evolution time, |∇ where  is the contrast parameter.Although the YK model has been demonstrated to be capable of avoiding staircase effect, isolated speckles will be produced in the evolution process.Besides, the high-frequency components are also oversmoothed, which leads to losing the edge and detail characteristics of the image.Based on the YK model, Hajiaboli introduced the gradient modulus in place of the absolute value of Laplacian as the edge indicator in the diffusion coefficient, and the improved fourth-order PDE model [18] is given by where Here |∇| is the gradient modulus which is used to detect the characteristics of the image .It has been proved that (7) can remove noise without bringing about isolated speckles at the cost of weakening edges.In addition to this, taking into the edge orientation, Hajiaboli proposed the following nonlinear fourth-order diffusion equation [19]: where  denotes the gradient direction and  is perpendicular to , that is, the direction of level set.  and   are the second-order directional derivatives of the image  in  and , respectively.Due to the anisotropic diffusion that the diffusion coefficient is smaller in the gradient direction than in the direction of level set, this denoising method has a better edge-preserving capacity, compared with the method in the literature [18].However, because of the diffusion inconformity in the directions of gradient and level set, the denoised image has staircase artifacts in the flat region when it comes to the distorted image corrupted by a high level of noise.

The Proposed Algorithm
3.1.New Edge Indicator.In the literature [8], Catté et al. introduced the Gaussian smoothing kernel to regularize the PM model, which can effectively smooth noise and then avoid staircase effect.Thereinto, the gradient information is based on the intensity similarity of each single pixel.Similarly, the existing edge indicators, such as, the gradient modulus and the absolute value of Laplacian, also depend on the intensity similarity of each single pixel, regardless of the change of the adjacent pixels around the processed pixel.Motivated by the idea in [20], we propose a new edge indicator employing the intensity similarity of neighbor pixels, that is, patch similarity modulus.The patch similarity modulus (, ) at the point (, ) of the image  is defined as where Image patches can represent structure information, such as edges and textures, while the single pixel cannot represent structure information [20].Therefore, the patch similarity modulus based on image patches can well describe the characteristics of the image.In order to objectively analyze the performance of the patch similarity modulus, Figure 1 From Figure 2(a), we can observe that the patch similarity modulus can distinguish between noise and edges.As for the edge-preserving ability, the patch similarity modulus is relatively weaker than the gradient modulus and the absolute value of Laplacian on account of the smaller intensity values on the edges.
However, the patch similarity modulus has a better noise removal ability, compared with the gradient modulus and the absolute value of Laplacian, as seen in Figure 2(b).It can be seen that the intensity value of the absolute value of Laplacian changes dramatically with large peaks and troughs in the homogeneous region corrupted by noise.As a result, the YK model based on the absolute value of Laplacian is faced with the problem that a small threshold value  will leave the noise in the processed image and a large  value will oversmooth the details of the target object.Therefore, the YK model cannot achieve a good balance between noise removal and edge preservation.In contrast, the intensity value of the patch similarity modulus changes steadily without positive and negative pulses.Moreover, the fluctuation of the patch similarity modulus is the smallest among the three edge indicators.It reveals that the patch similarity modulus is robust to noise.Through the above analysis, it can be seen that the patch similarity modulus can effectively distinguish the different characteristics of the image, such as edges and noise.And the denoising algorithm based on the patch similarity modulus can effectively remove noise.

New Fourth-Order PDE.
In this paper, aiming at the shortages of the existing methods discussed in Section 2, we propose a new fourth-order PDE for image denoising.
Primarily, for removing noise effectively, we make full use of the advantage of the patch similarity modulus  and propose a new diffusion coefficient expression given by where  is the threshold value of .Therefore, the fourthorder PDE based on the patch similarity modulus is proposed as Then, for decreasing the diffusion coefficient of edges, the difference curvature [21] is introduced in the diffusion process to preferably preserve edge information.The difference curvature is formulated as where   and   represent the second derivatives in the direction normal and tangent to the level curves, respectively, and | ⋅ | denotes the absolute function.Table 1 shows the characteristic analysis of the difference curvature in the three different regions of the distorted image.From Table 1, we can see that the difference curvature  is only large on edge features and small in the homogeneous and noisy regions.
Obviously, edges can be distinguished from noisy and flat regions by the value of .To have the same monotonicity as () with respect to , the function () concerning  is defined as Here, () is in inverse proportion to , which ensures that () and () are monotonically decreasing functions.() is bounded in (0, 1]. All the facts discussed above motivated us to combine () and () to determine the diffusion process.Therefore, the new fourth-order PDE based on the patch similarity modulus and the difference curvature for image denoising is proposed as follows: According to the analysis above, the performance of the proposed fourth-order PDE is described as follows.For noisy pixels, a large () value is chosen for the small value of the patch similarity modulus  and the () value tends to be 1 for the small difference curvature .Then, the diffusion coefficient () ⋅ () is approximately equal to (), which will accelerate the diffusion process of noisy pixels.Therefore, the proposed algorithm can effectively filter out noise.Since the intensity value of  at the noisy pixel is very small and changes smoothly in a small scope, a small threshold value  can suppress noise, which will preserve more details at the same time.Aiming at edges, the value of  is very large.Then, the () value tends to be 0. Consequently, the diffusion coefficient () ⋅ () is approximately equal to 0, which will slow down the diffusion process of edges to preserve the characteristics of the image.To sum up, the proposed algorithm can adaptively adjust the diffusion speed of edges, details, and noise in the distorted image and has a strong capability of removing noise and preserving edges and details.
To justify the contribution of the patch similarity modulus and the difference curvature, we conduct an experiment as shown in Figure 3. Figures 3(c) and 3(d) show the comparative results processed by ( 13) and ( 16), respectively.From Figure 3(c), we can see that the fourth-order PDE based on the patch similarity modulus can effectively remove noise.But this method obscures some edges.From Figure 3(d), we can see that the fourth-order PDE based on the patch similarity modulus and the difference curvature can not only remove noise, but also preserve edges.This means that the patch similarity modulus plays a crucial role in suppressing noise and the difference curvature is mainly used to preserve edges.
For demonstrating the convergence of the proposed algorithm, the "normalized step difference energy" (NSDE) [22] is calculated at each iteration: where   and  −1 denote the image at  and  − 1 iteration, respectively.Figure 4 shows the NSDE graph of the proposed algorithm for Figure 3(b).We can see that the NSDE graph decreases with the increase of the number of iterations, which shows that the proposed algorithm has a convergence.

Numerical Implementation
We employ the explicit Euler numerical scheme for solving the PDE given in (16).Assuming that Δ is the iterative time step and ℎ is the space grid size, we quantize the time and space coordinates as follows: = Δ,  = 0, 1, 2, 3, . . .,   = ℎ,  = 0, 1, 2, . . ., ,   = ℎ,  = 0, 1, 2, . . ., . ( ℎ × ℎ is the size of the image support.In the paper, we use the grid size ℎ = 1; thus, the Laplacian at the point (, ) of the image  after  iterations is calculated by Let   , = (  , ) ⋅ (  , ) ⋅ ∇ 2   , ; then the Laplacian of   , can be calculated by with symmetric boundary conditions: Then, ( 16) can be discretized as In addition, the difference curvature   at  iteration is calculated by where Here, the central difference scheme is used for the spatial derivatives above; that is, . (26)

Experimental Results and Analysis
We have used two 256 × 256 images with both textured and smooth regions, namely, Lena and Barbara, to test the performance of our algorithm.The original images are degraded by Gaussian noise with the mean value of zero and standard deviation of  = 10,  = 15, and  = 20, respectively.For the sake of demonstrating the accuracy and superiority of the proposed algorithm, related denoising methods, including the PM model in [7], the YK model in [12], the self-governing hybrid (SGH) model in [18], and the anisotropic fourth-order diffusion (AFOD) filter in [19], are carried out for comparative experiments.Moreover, the mean absolute error (MAE), peak signal to noise ratio (PSNR), and mean structural similarity (MSSIM) [23] are adopted to evaluate the denoising quality objectively.The time step is chosen as Δ = 0.03 for the SGH model, the AFOD filter, and the proposed algorithm [19], because these diffusion methods are highly sensitive to the time step.In the PM model and the YK model, the time step is chosen as Δ = 0.25.For the proposed algorithm, to evaluate the effect of the edge threshold parameter , Figures 5 and 6 present the denoising results under varying values of  on the Lena image corrupted by Gaussian noise with the standard deviation of 10 and 15, respectively.The denoising results in Figures 5 and 6 are obtained at a fixed number of iterations  = 350 and  = 500, respectively.Figures 5(b)-5(d) show that  = 2 is the optimal parameter value that gives the best visual diffusion result when  = 10, while  = 3 is Mathematical Problems in Engineering the optimal parameter value when  = 15, as can be seen from Figures 6(b)-6(d).The other two ( = 2 and 4) are either too small or too large, which will lead to leaving noise in the denoised image or oversmoothing the image details.Therefore, the edge threshold  needs to be increased for the larger standard deviation of noise.The number of iterations  has been hand-tuned to produce the good result for different images.
5.1.Image Quality Measures.Let  0 be the original image and let  be the denoised image, and assume  ×  to be the size of  0 .The MAE and PSNR are, respectively, defined as Here, the smaller the MAE value is, the better the denoising effect will be.And the larger the PSNR value is, the less the image distortion will be.
Apart from the above two measures, the MSSIM index [23] is used to evaluate the overall image quality, defined as where   and   denote the content of the th local window in original and denoised images, respectively,  is the number of local windows of the image, and the structural similarity SSIM(  ,   ) is given by where    and    denote the mean value of   and   , respectively,    and    denote the variance of   and   , respectively,      is the covariance of   and   , and 2 , where  is the dynamic range of pixels values (255 for 8-bit grayscale images) and  1 = 0.01 and  2 = 0.03 are constants.
Here, the MSSIM can measure the similarity of two images including the luminance, contrast, and structure and a larger MSSIM value indicates a better image structurepreserving capacity.7(c)-7(g).In order to clearly display the denoising effects of the related methods, Figure 8 presents the enlarged portions of the images shown in Figure 7. From Figure 8(c), it can be observed that the PM model creates evident staircase effect in the denoised image.Although the YK model suppresses staircase effect, this model creates isolated speckles, as seen in Figure 8(d).Figures 8(e) and 8(l) illustrate that the SGH model loses some details, in spite of the good denoising ability.From Figure 8(f), we can observe that the denoised image obtained by the AFOD      On the whole, compared with the results obtained from the related methods, the denoised image of the proposed algorithm has a better visual effect and is more similar to the original image, which demonstrates that the proposed algorithm is able to get the best denoising performance.

Comparative Results.
In order to quantitatively analyze the performance of our algorithm, the MAE, PSNR, and MSSIM values of the Lena and Barbara images processed by various denoising methods at different noise levels are summarized in Tables 2 and  3, respectively.According to the MAE, PSNR, and MSSIM values of the denoised images, it is obvious that the proposed algorithm outperforms the other methods.Especially for the Barbara image, the PSNR values obtained by the proposed algorithm are obviously higher than those obtained from the other methods, which further illustrates that the proposed algorithm has a better superiority when the noisy image contains more texture details.Based on the above visual and quantitative analysis, the experimental results demonstrate that the proposed algorithm, which is more suitable to deal with the noisy images with abundant details, can preserve the structural features of the original image and obtain the highquality denoised image.

X-Ray Image Denoising.
In the field of X-ray nondestructive testing, X-ray image denoising is a very critical premise for precisely recognizing defects.Figures 11 and 12 show the denoising results of two printed circuit board (PCB) X-ray images.For Figure 11(a), the parameters of the proposed algorithm are as follows: 150 iterations,  = 2 and  = 0.03.And Figure 12(b) is obtained with  = 4 and  = 0.03 after 450 iterations.Figures 11(b) and 12(b) illustrate that the proposed algorithm can well remove random noise without damaging edges and details.Therefore, the proposed algorithm has a promising application in the X-ray image denoising.

Conclusions
In this paper, on the basis of analyzing the shortages of the PM model, the YK model, the SGH model, and the AFOD filter, we have proposed a new fourth-order partial differential equation for image denoising.In the proposed algorithm, the patch similarity modulus which can suppress noise and the difference curvature which can preserve edges are used to determine the diffusion coefficient of edges, details, and noise.And the denoising theory of the proposed algorithm has been described in detail.Besides, the accuracy and versatility of the proposed algorithm have been verified by variedly noisy images.Experimental results have demonstrated that (a) presents a geometry image, and Figure 1(b) is generated by adding Gaussian noise with the mean value of zero and standard deviation of  = 15 to Figure 1(a).Figures 2(a) and 2(b) plot the intensity values of the gradient modulus, the absolute value of Laplacian, and the patch similarity modulus along Line 1 via the edge and along Line 2 within the flat region in Figure 1(b), respectively.

Figure 7 (
a) is the original Lena image and Figure 7(b) presents the noisy image corrupted by Gaussian noise with the standard deviation of  = 10.The edge threshold  is set to 2 in all compared methods.The results of various denoising methods and the corresponding number of iterations are shown in Figures

Table 1 :
The characteristic analysis of difference curvature.

Table 2 :
The MAE, PSNR, and MSSIM values of various methods for Lena image at different noise levels.

Table 3 :
The MAE, PSNR, and MSSIM values of various methods for Barbara image at different noise levels.